Trapped-ion Instability Research Articles

Trapped-ion driven turbulence in tokamak plasmas

Ion turbulence is expected to play an important role in anomalous transport. Thus an original approach is proposed here to study trapped-ion instability. A Vlasov code is used to determine the behaviour of the instability near the threshold and compared with analytical solutions of the Vlasov equation. Some interesting features which appear in the nonlinear regime are discussed.

Collisionless Trapped Ion Temperature Gradient Instabilities11The project supported by Chinese Nuclear Science Foundation under Contract No. 192010

Under the fluid limit, the collisionless electrostatic trapped ion temperature gradient instabilities are investigated by retaining the trapped ion parallel compressibility in the long wavelength limit. The two-scale analysis should be employed in view of the fact that the eigenfunctions vary over the connection width scale with an envelope varying over a secular scale. The results show that the slab branch of trapped ion modes, corresponding to the aperiodic trapped ion instability discovered by Kadomtzev, propagates along ion diamagnetic direction. The toroidd branch is also obtained in low-frequency and long wavelength limit. In this case, besides the unstable one propagating in ion diamagnetic direction, there exists a marginally stable toroidal odd mode propagating in electron diamagnetic direction. The eigenfrequencies and perturbative structures of these unstable modes are given analytically and numerically. The numerical results are in good agreement with the analytical ones.

On the stability of the internal kink mode in the banana regime

The stability of the internal kink mode is investigated taking into account the kinetic response associated to the trapped thermal ions. Ion–ion collisions and diamagnetic effects in the layer are also considered. A significant stabilizing contribution is obtained, even at low-β values, on the mode, which might be stable, on present experiments, even though predicted unstable according to the Bussac criterion [Bussac et al., Phys. Rev. Lett. 35, 1638 (1975)]. In addition, a trapped-ion instability is found, characterized by mode frequency of the order of the trapped-ion bounce-averaged magnetic drift frequency.

Long-wavelength microinstabilities in toroidal plasmas*

Realistic kinetic toroidal eigenmode calculations have been carried out to support a proper assessment of the influence of long-wavelength microturbulence on transport in tokamak plasmas. In order to efficiently evaluate large-scale kinetic behavior extending over many rational surfaces, significant improvements have been made to a toroidal finite element code used to analyze the fully two-dimensional (r,θ) mode structures of trapped-ion and toroidal ion temperature gradient (ITG) instabilities. It is found that even at very long wavelengths, these eigenmodes exhibit a strong ballooning character with the associated radial structure relatively insensitive to ion Landau damping at the rational surfaces. In contrast to the long-accepted picture that the radial extent of trapped-ion instabilities is characterized by the ion-gyroradius-scale associated with strong localization between adjacent rational surfaces, present results demonstrate that under realistic conditions, the actual scale is governed by the large-scale variations in the equilibrium gradients. Applications to recent measurements of fluctuation properties in Tokamak Fusion Test Reactor (TFTR) [Plasma Phys. Controlled Nucl. Fusion Res. (International Atomic Energy Agency, Vienna, 1985), Vol. 1, p. 29] L-mode plasmas indicate that the theoretical trends appear consistent with spectral characteristics as well as rough heuristic estimates of the transport level. Benchmarking calculations in support of the development of a three-dimensional toroidal gyrokinetic code indicate reasonable agreement with respect to both the properties of the eigenfunctions and the magnitude of the eigenvalues during the linear phase of the simulations of toroidal ITG instabilities.

Trapped particle dynamics in toroidally rotating plasmas

A detailed single particle orbit analysis in toroidally rotating plasma yields new analytical formulas for the second adiabatic invariant, the bounce frequency, and the precession frequency up to the first-order correction in ρpi (poloidal ion gyroradius)/Lv (scale length of rotation velocity), for toroidal flow values of the order of ion thermal velocity. Toroidal plasma rotation effects on the trapped ion instabilities in tokamaks are investigated in the context of local theory. Toroidal plasma rotation increases both the fraction of trapped particles and their precession drift velocity. Consequently, the growth rate of trapped ion instability increases in both dissipative and collisionless regimes.

Growth rate, saturation, and radial transport measurements for the trapped ion instability

The dissipative trapped ion instability (DTII) is observed as an m=1 mode localized to the mirror cell of the Columbia Linear Machine (CLM) [Plasma Phys. 24, 185 (1982)]. A gated feedback diagnostic allows the direct measurement of the DTII linear growth rate. The DTII growth rate is measured as a function of trapped fraction, keeping all other parameters fixed, and clearly displays the scaling with trapped fraction predicted by the radially local linear dispersion relation, including ion Landau damping. The saturated DTII mode amplitude is measured concurrently with the growth rate. The saturated amplitude displays a nearly linear dependence on the measured growth rate, normalized to the DTII real frequency, and is nearly proportional to the square root of the measured growth rate. A striped particle collector is used to measure the ambipolar radial plasma flux as the trapped fraction is varied. Radial diffusion coefficients are calculated from the measured radial particle fluxes and display a scaling intermediate between the Kadomtsev weak and strong turbulence predictions.

Bounce Time Averaged Wave Potential and the Dissipative Trapped Ion Instability

Effect of banana orbit of the trapped particle on the dissipative trapped ion instability (DTII) is investigated. As the frequency of the DTII is much smaller than the bounce frequency, wave potential is averaged over the bounce period or along the banana orbit of the trapped particle. With this averaged wave potential, dispersion relation is obtained including the effect of the banana orbit. Stabilizing effect is not so large as expected. The growth rate is positive for the wide range of the poloidal mode number m . Previous results, based on the growth rate which makes use of an analogy to the finite Larmor radius effect, overestimate the damping by the effect of the banana orbit incorrectly.

Nonlinear saturation of single-mode trapped ion instability

The trapped ion instability appears as a single mode in the Columbia Linear Machine and is expected to appear as such in tandem mirrors. Using three-wave, nonresonant mode coupling of radially nonlocal eigenfunctions, two mode-coupling equations for the unstable fundamental and stable first-order radial harmonics are derived and solved. Mode coupling to the damped higher-order radial harmonics is the principal physical saturation mechanism of the unstable fundamental mode.

Experimental study of the dissipative trapped ion instability

A new, steady-state linear machine has been constructed which produces a low-density, low-temperature hydrogen plasma in the collisionality regime of the dissipative trapped ion instability. This instability was seen and identified through the observed dependence of the wave amplitude on trapped fraction, axial and radial position, and collisionality and through the scaling of the real frequency with trapped fraction. The observed radial and axial mode structure of the instability is compared with the results of a simple radial model and a calculation of the Landau damping for both trapped and transit particles in a linear geometry.

Numerical Investigation of Saturation Mechanism of the Dissipative Trapped Ion Instability II

Expressions of the anomalous diffusion coefficient due to the dissipative trapped ion instability (DTII) are derived for the cases with and without the effect of magnetic shear. Derivation is made by taking into account of the single mode saturation of the DTII previously obtained numerically. In the absence of the shear effect, the diffusion coefficient is proportional to ν i a 2 (ν i is the effective collision frequency of the trapped ions and a is the minor radius of a torus) and is much larger than the neoclassical ion thermal conductivity. In the presence of the shear effect, the diffusion coefficient is much smaller than Kadomtsev and Pogutse's value and is the same order of magnitude as the neoclassical ion thermal conductivity.

Saturation of trapped-ion instability by two-dimensional electrostatic trapping

The non-linear evolution of trapped-ion instability in Tokamaks is investigated. The authors first derive a kinetic model of mode coupling, but show that it cannot be readily applied to resonant curvature-driven modes. They then show that non-linear saturation can result from trapping of resonant particles in the electrostatic potential of the wave. This mechanism is strongly enhanced by the radial structure of the mode, resulting in very low amplitudes at saturation.

A steady state linear machine for collisionless plasma studies

The authors describe a new linear, steady state machine for the production of a quiescent, magnetically confined, collisionless hydrogen plasma. This new device is particularly well suited for the study of low frequency drift wave type instabilities such as the trapped ion instability. This represents an extension of the parameter space accessible to linear devices and more closely simulates the collisionality and bounce frequency regimes ( nu / omega b<or approximately=0.1) of a thermonuclear plasma.

Experimental Study of the Dissipative Trapped Ion Instability

Experiments have been performed in a steady-state linear machine with a low temperature, low density plasma in the collisionality regime of the dissipative trapped ion instability. This instability was seen and identified through the observed dependence of wave amplitude on trapped fraction, axial and radial structure, and collisionality and through the scaling of the real frequency with trapped fraction.

Numerical Investigation of Saturation Mechanism of the Dissipative Trapped Ion Instability

A set of model equations of the dissipative trapped ion instability is derived and solved numerically. Multiple time scale expansion scheme is applied to the non-linear fluid equations of Kadomtsev and Pogutse by taking µ=ω/ν e as a small parameter, where ω is the frequency of the wave and ν e is the effective collision frequency of the electrons. All the modes except for the linearly most unstable one die away due to the mode coupling. Final saturation of the mode is attributed to the modification of background trapped ion number density averaged over the poloidal direction. The coefficient of trapped ion anomalous diffusion, which scales as D ∝ν i a 2 ( ν i is the effective collision frequency of the ions and a is the minor radius of a torus) is also derived.

Production and Observation of the Dissipative Trapped-Ion Instability

A new, steady-state linear mirror machine has been constructed which produces a low-density, low-temperature hydrogen plasma in the collisionality regime of the dissipative trapped-ion instability. This instability was seen and identified through the observed dependence of wave amplitude on trapped fraction, axial and radial position, and collisionality and through the scaling of real frequency with trapped fraction.

Mixing Length Hypothesis in the Dissipative Trapped Ion Instability

The mixing length hypothesis is investigated for the purpose of making the underlying idea of the hypothesis clear in the case of the dissipative trapped ion instability. Three different approaches are employed; 1) analogy to the ordinary turbulence in hydrodynamics, 2) a nonlinear turbulent collision theory, and 3) a quasilinear theory of the density flattening. If the isotropic turbulence is assumed, the mixing length hypothesis can be derived by the first two approaches. The formula D ≃γ/ k ⊥ 2 , which is usually obtained from the mixing length hypothesis, can be derived by the third approach.

Effects of Impurity Ions on the Anomalous Diffiusion due to the Dissipative Trapped Ion Instability

Anomalous diffusion coefficient is evaluated by using result of the renormalized perturbation theory for the turbulent collision. Two cases are investigated for light impurity ions, one in the plateau regime and the other in the banana regime. Time evolution of the wave speetrum is solved numerically and the diffusion coefficient is determined, by taking the density gradient of impurity ions in the same direction as that of the bulk plasma. Even with a small concentration of the impurity the diffusion coefficient takes a smaller value for both regime cases.

New scaling laws for trapped-particle anomalous transport in tokamaks

An improved formula for anomalous transport caused by the dissipative trapped-ion instability (without shear effect) is derived numerically and by two independent analytical methods. The new shearless result is also used to derive the anomalous diffusion with shear effect by a general method already published.

Two-dimensional analysis of trapped-ion eigenmodes

The first fully two-dimensional eigenmode analysis of the trapped-ion instability in an axisymmetric toroidal geometry is presented. The poloidal structure is taken into account by Fourier-expanding the perturbed electrostatic potential, φ, in ϑ. Assuming that the perturbation varies mildly over a typical ion-banana-width, ρbi, each poloidal harmonic is expressed as a truncated Taylor series in the minor radius to account for the radial structure. The resulting set of coupled ordinary second-order differential equations is solved numerically by the method of finite elements. The formalism is also applicable to the radially local and one-dimensional radial approximations. Results obtained in these limits are presented and found to be in reasonable agreement with previous calculations. In low shear plasmas, the full two-dimensional calculation is in qualitative agreement with the one-dimensional radial approximation. However, for higher shear the two-dimensional calculation yields a somewhat different picture for the radial structure of the instability. The analysis presented is limited to long radial wavelength and electrostatic perturbations.

Scaling law for trapped-ion anomalous diffusion in tokamaks with shear

A 1/B3 scaling law is derived for anomalous diffusion produced by the dissipative trapped-ion instability in axisymmetric toroidal plasmas with shear. The theory presented by the authors helps to explain why transport of this type has not been detected in the PLT heating experiment, so far. In addition, it is shown that the Kadomtsev-Pogutse diffusion formula, will contradict the trapped-fluid equations from which it was deduced, provided that the necessary boundary conditions are observed.